Integration Practice Questions
AP (Advanced Placement) · AP Calculus BC · 140 free MCQs with instant results and detailed explanations.
140
Total
32
Easy
80
Medium
28
Hard
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Sample Questions from Integration
Here are 10 sample questions. Start a quiz to get randomized questions with scoring.
Q1
Easy
Evaluate the integral ∫ (3x^2 + 2) dx from 1 to 3.
A.
40
B.
34
C.
25
D.
30
Show Answer & Explanation
Correct Answer: B
The definite integral evaluates to 34 by finding the antiderivative and computing the difference between F(3) and F(1).
Q2
Easy
Which of the following represents the area under the curve y = x^2 from x = 0 to x = 2?
A.
2/3
B.
2
C.
8/3
D.
4/3
Show Answer & Explanation
Correct Answer: C
The area under the curve is found by evaluating the definite integral of x^2 from 0 to 2, resulting in 8/3.
Q3
Easy
What is the integral of the function f(x) = 3x^2 with respect to x?
A.
x^3 + C
B.
3x^3 + C
C.
x^2 + C
D.
3x + C
Show Answer & Explanation
Correct Answer: A
The integral of 3x^2 is found using the power rule for integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C. Applying this rule gives us (3x^3)/3 + C = x^3 + C.
Q4
Medium
What is the integral of the function f(x) = 3x^2 + 2x - 5 with respect to x?
A.
x^3 + x^2 - 5x + C
B.
x^3 + x^2 - 5
C.
x^3 + 2x^2 - 5 + C
D.
3x^3 + x^2 - 5 + C
Show Answer & Explanation
Correct Answer: A
The integral of f(x) = 3x^2 + 2x - 5 is computed by applying the power rule for integration. Each term is integrated separately, resulting in x^3 + x^2 - 5x + C.
Q5
Medium
Evaluate the definite integral from 1 to 4 of the function f(x) = 2x + 3.
A.
26
B.
31
C.
24
D.
21
Show Answer & Explanation
Correct Answer: A
The definite integral from 1 to 4 of f(x) = 2x + 3 is calculated by finding the antiderivative, which is x^2 + 3x. Evaluating from 1 to 4 gives (16 + 12) - (1 + 3) = 26.
Q6
Medium
Find the value of the integral ∫(sin(x)cos(x) dx).
A.
0.5sin^2(x) + C
B.
sin^2(x) + C
C.
-0.5sin^2(x) + C
D.
-sin(x)cos(x) + C
Show Answer & Explanation
Correct Answer: A
The integral ∫(sin(x)cos(x) dx) can be simplified using the substitution sin^2(x) = (1/2)sin(2x). The result gives 0.5sin^2(x) + C.
Q7
Medium
Determine the integral of f(x) = e^(2x) from x = 0 to x = 1.
A.
e^2 - 1
B.
1 - e^2
C.
2(e - 1)
D.
e - 1
Show Answer & Explanation
Correct Answer: A
The integral of f(x) = e^(2x) can be found using substitution. The result is (1/2)e^(2x) evaluated from 0 to 1, which calculates to e^2 - 1.
Q8
Hard
Evaluate the integral \( \int_0^1 (x^3 - 2x^2 + x) e^{x^2} \, dx \).
A.
\( \frac{1}{2} e - 1 \)
B.
\( \frac{1}{2} e - \frac{3}{4} \)
C.
\( \frac{1}{2} e - \frac{1}{2} \)
D.
\( 0 \)
Show Answer & Explanation
Correct Answer: B
Using integration by parts, we can evaluate the integral with \( u = (x^3 - 2x^2 + x) \) and \( dv = e^{x^2} dx \). After performing the integration and substituting the limits, we find that the result simplifies to \( \frac{1}{2} e - \frac{3}{4} \).
Q9
Hard
Find the area enclosed by the curves \( y = \sin(x) \) and \( y = \sin(2x) \) from \( x = 0 \) to \( x = \pi \).
A.
\( 1 \)
B.
\( \frac{1}{2} \)
C.
\( \frac{3}{4} \)
D.
\( \frac{1}{4} \)
Show Answer & Explanation
Correct Answer: A
To find the area enclosed by the curves, we first determine their points of intersection within the given interval, then set up the integral of the absolute difference of the two functions. The area turns out to be equal to 1 after evaluating the integral.
Q10
Hard
Evaluate the integral \( \int_0^1 (x^3 - 3x^2 + 3x) e^{x^3} \, dx \).
A.
\( e - 1 \)
B.
\( e - 3 \)
C.
\( 1 \)
D.
\( 0 \)
Show Answer & Explanation
Correct Answer: A
The integral can be solved using integration by parts, where you let \( u = x^3 - 3x^2 + 3x \) and \( dv = e^{x^3} dx \). The result becomes \( e^{x^3} (x^3 - 3x^2 + 3x) \) evaluated from 0 to 1, simplifying to \( e - 1 \).
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Integration — AP (Advanced Placement) AP Calculus BC Practice Questions Online
This page contains 140 practice MCQs for the chapter Integration in AP (Advanced Placement) AP Calculus BC. The questions are organized by difficulty — 32 easy, 80 medium, 28 hard — so you can choose the right level for your preparation.
Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.